On the distribution of $\alpha p^2$ modulo one over primes of the form $[n^c]$

Let $[\,\cdot\,]$ be the floor function and $\|x\|$ denotes the distance from $x$ to the nearest integer. In this paper we show that whenever $\alpha$ is irrational and $\beta$ is real then for any fixed $\frac{13}{14}<\gamma<1$, there exist infinitely many prime numbers $p$ satisfying the inequality \begin{equation*} \|\alpha p^2+\beta\|< p^{\frac{13-14\gamma}{29}+\varepsilon}\end{equation*} and such that $p=[n^{1/\gamma}]$.